Appears in collection : 2023 IHES Summer School – Recent Advances in Algebraic K-theory

It was a fundamental insight by Thomason (building on work of Waldhausen) that algebraic K-theory of a ring or scheme could be defined via the category of perfect complexes and that K-theory sends a Verdier quotient sequence of such categories to a fiber sequence of spectra. In modern terms: K-theory is a localizing invariant. This in particular yields descent properties, e.g. Nisnevich descent, for K-theory or more generally any localizing invariant. I plan to give an introduction to these by now classical topics and discuss recent developments and applications, in particular for algebraic K-theory.